Q27 / 75 · JEE Main 2025 (3 Apr Shift 1) · Mathematics

The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is

  • A
    $16923$
  • B
    $3763$
  • C
    $33845$
  • Correct answer
    D
    $18817$

Solution

$S=(2+\sqrt{3})^8$ For sum of rational terms $\begin{aligned} & ={ }^8 \mathrm{C}_0(2)^8+{ }^8 \mathrm{C}_2(2)^6 \cdot(\sqrt{3})^2+{ }^8 \mathrm{C}_4(2)^4(\sqrt{3})^4 \\ & \quad+\quad{ }^8 \mathrm{C}_6(2)^2(\sqrt{3})^6+{ }^8 \mathrm{C}_8(\sqrt{3})^8 \\ & =2^8+28 \times 2^6 \cdot 3+70.2^4 \cdot 9+28.2^2 .27+81 \\ & =256+5376+10080+3024+81 \\ & =18817\end{aligned}$

Hence, the correct answer is option: 'D'.

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